🔗 Algebra Solver

Compound Inequality Calculator - AND & OR

Q: What is a compound inequality?

A compound inequality combines two inequalities using AND or OR. AND means both conditions must be true, while OR means at least one condition must be true.

Q: How do you solve a compound inequality?

Isolate the variable by applying inverse operations. For a three-part inequality, apply the same operation to all three parts. If you multiply or divide by a negative number, reverse the inequality signs.

Q: What is the difference between AND and OR inequalities?

An AND inequality uses the intersection of two solution sets, meaning both conditions must be satisfied. An OR inequality uses the union, meaning either condition can be satisfied.

Q: How do you graph a compound inequality?

Graph the boundary values on a number line. Use open circles for strict inequalities and closed circles for inclusive inequalities. Shade the region that satisfies the AND or OR conditions.

Q: How is a compound inequality written in interval notation?

Use square brackets when an endpoint is included and parentheses when it is excluded. For example, -1 ≤ x ≤ 6 is written as [-1, 6], while x < 3 is written as (-∞, 3).

Solve compound inequalities with step-by-step solutions, AND/OR conditions, graphs, and interval notation.

Compound Inequality

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How to Use & Educational Guide

In-depth guide for Compound Inequality Calculator

Educational Overview

What Is a Compound Inequality?

A compound inequality combines two inequalities using AND or OR. An AND inequality requires both conditions to be true, so its solution is the intersection of the two solution sets. An OR inequality requires at least one condition to be true, so its solution is the union of the two sets.

Compound inequalities can be written as a three-part inequality such as $2 \le x + 3 \le 9$, or as two separate inequalities such as $x < 2$ OR $x \ge 5$. The solution can be expressed as an inequality, on a number line, or in interval notation.

Compound Inequality Rules

The method used to solve a compound inequality depends on whether the conditions are connected by AND or OR.

AND: $a < x < b$ → values between both boundaries

OR: $x < a \;\text{ OR }\; x > b$ → values outside the boundaries

Negative multiplication/division → flip the inequality sign

In interval notation, use square brackets for included endpoints and parentheses for excluded endpoints. Infinity always uses parentheses.

How to Solve a Compound Inequality Step-by-Step?

Use these steps to solve most linear compound inequalities:

Identify the type: Determine whether the problem uses AND or OR.
Isolate the variable: Apply addition, subtraction, multiplication, or division to solve each inequality.
Apply operations correctly: For a three-part AND inequality, perform the same operation on all three parts.
Flip the sign when necessary: Reverse the inequality direction whenever you multiply or divide by a negative number.
Combine the solution: For AND, keep the values that satisfy both conditions. For OR, include values satisfying either condition.
Write interval notation: Convert the final solution into interval notation when needed.

Compound Inequality Examples

Example 1: Solve an AND Compound Inequality: $2 \le x + 3 \le 9$

• Step 1: Subtract 3 from all three parts: $2 - 3 \le x + 3 - 3 \le 9 - 3$.

• Step 2: Simplify: $-1 \le x \le 6$.

• Step 3: Convert to interval notation: $[-1, 6]$.

👉 Final Solution: $-1 \le x \le 6$

Example 2: Solve a Compound Inequality with a Negative Coefficient: $-4 < -2x + 2 \le 8$

• Step 1: Subtract 2 from all three parts: $-6 < -2x \le 6$.

• Step 2: Divide all three parts by $-2$ and flip both inequality signs: $3 > x \ge -3$.

• Step 3: Rewrite in ascending order: $-3 \le x < 3$.

• Step 4: Convert to interval notation: $[-3, 3)$.

👉 Final Solution: $-3 \le x < 3$

Example 3: Solve an OR Compound Inequality: $2x - 1 < 5$ OR $3x + 2 \ge 17$

• First inequality: $2x - 1 < 5 \implies 2x < 6 \implies x < 3$.

• Second inequality: $3x + 2 \ge 17 \implies 3x \ge 15 \implies x \ge 5$.

• Combine with OR: $x < 3$ OR $x \ge 5$.

• Interval notation: $(-\infty, 3) \cup [5, \infty)$.

👉 Final Solution: $(-\infty, 3) \cup [5, \infty)$

How to Graph a Compound Inequality?

Compound inequalities can be shown on a number line to make the solution range easier to understand. Use an open circle when the boundary is not included ($<$ or $>$) and a closed circle when the boundary is included ($\le$ or $\ge$).

AND inequality: Shade the region where both conditions overlap.
OR inequality: Shade both regions that satisfy either condition.
Infinite endpoint: Use an arrow to show that the solution continues indefinitely.

Real-World Applications of Compound Inequalities

📏 Measurement Tolerances

Manufacturing specifications often require a measurement to stay within a permitted range. For example, a part with a tolerance of $9.8$ mm to $10.2$ mm can be represented as $9.8 \le x \le 10.2$.

🌡️ Safe Operating Ranges

Temperature, speed, pressure, and other operating limits can be represented with compound inequalities when a value must remain within a specified range.

Common Mistakes to Avoid

Forgetting to flip the signs: Always reverse the inequality direction when multiplying or dividing by a negative number.
Confusing AND with OR: AND requires both conditions to be true, while OR requires at least one condition to be true.
Using the wrong interval brackets: Use square brackets for included endpoints and parentheses for excluded endpoints.
Applying an operation to only part of a three-part inequality: When solving $a < bx + c \le d$, apply the same valid operation to all three parts.

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Frequently Asked Questions (FAQs)

Answers to common questions about this calculator

What is a compound inequality?

A compound inequality combines two inequalities using AND or OR. AND means both conditions must be true, while OR means at least one condition must be true.

How do you solve a compound inequality?

Isolate the variable by applying inverse operations. For a three-part inequality, apply the same operation to all three parts. If you multiply or divide by a negative number, reverse the inequality signs.

What is the difference between AND and OR inequalities?

An AND inequality uses the intersection of two solution sets, meaning both conditions must be satisfied. An OR inequality uses the union, meaning either condition can be satisfied.

How do you graph a compound inequality?

Graph the boundary values on a number line. Use open circles for strict inequalities and closed circles for inclusive inequalities. Shade the region that satisfies the AND or OR conditions.

How is a compound inequality written in interval notation?

Use square brackets when an endpoint is included and parentheses when it is excluded. For example, -1 ≤ x ≤ 6 is written as [-1, 6], while x < 3 is written as (-∞, 3).